Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} n !\left(\frac{x}{10}\right)^{n}$$

Short answer

Radius of convergence is 0; interval of convergence is [0, 0].

Step-by-step solution

Identify the power series

The given power series is \(\sum_{n=0}^{\infty} n !\left(\frac{x}{10}\right)^{n}\). Here the general term is given by \(a_n = n!\left(\frac{x}{10}\right)^{n}\).

Use the Ratio Test for radius of convergence

To find the radius of convergence, we will apply the Ratio Test. The Ratio Test states that a series \(\sum a_n\) converges if \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1\). For our series, the terms are \(a_n = n!\left(\frac{x}{10}\right)^{n}\) and \(a_{n+1} = (n+1)!\left(\frac{x}{10}\right)^{n+1}\).

Calculate the ratio \( \frac{a_{n+1}}{a_n} \)

Calculate the next term over the current term: \[\frac{a_{n+1}}{a_n} = \frac{(n+1)!\left(\frac{x}{10}\right)^{n+1}}{n!\left(\frac{x}{10}\right)^{n}} = \frac{(n+1) \cdot x}{10}.\]

Take the limit of the ratio

Evaluate the limit: \[\lim_{n \to \infty} \left| \frac{(n+1) \cdot x}{10} \right| = \lim_{n \to \infty} \left| \frac{n+1}{10} \right| \cdot |x| = \infty\]for any nonzero \(x\). This implies that the series diverges for any \(x eq 0\).

Determine the radius of convergence

Since the limit goes to infinity for any non-zero \(x\), the series only converges when \(x=0\). Therefore, the radius of convergence \(R\) is 0.

State the interval of convergence

The interval of convergence is simply the closed interval where the series converges. Since the series only converges at \(x = 0\), the interval of convergence is the single point \([0, 0]\).

Key concepts

Power Series

A power series is a type of infinite series that involves powers of a variable, typically x. It is written in the form:

• \( \sum_{n=0}^{\infty} c_n (x - a)^n \)

Here, \( c_n \) represents the coefficients, \( a \) is the center of the series, and \( x \) is the variable.
Power series are useful in representing many different functions, including polynomial approximations of transcendental functions like \( e^x \), \( \sin x \), and \( \cos x \).
The series allows us to analyze and understand complex functions by studying their simpler polynomial forms. In our exercise, the given power series is \( \sum_{n=0}^{\infty} n! \left(\frac{x}{10}\right)^n \). This shows the series with numeric coefficients dependent on \( n! \), factorial of \( n \), rather than a constant coefficient \( c_n \).

Ratio Test

The Ratio Test is a powerful tool used to determine the convergence or divergence of an infinite series. For a series \( \sum a_n \), the test compares successive terms to see if they tend towards zero quickly enough.
The test is applied by considering the limit:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

If the limit is less than 1, the series converges.
If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive, and further analysis is needed.
In our problem, using the Ratio Test on \( \sum_{n=0}^{\infty} n! \left(\frac{x}{10}\right)^n \) results in \( \lim_{n \to \infty} \left| \frac{(n+1) \cdot x}{10} \right| \), which diverges for any \( x eq 0 \), indicating a radius of convergence of 0.

Interval of Convergence

The interval of convergence of a power series is the set of all \( x \) values for which the series converges. For a convergent series, this interval is centered at \( a \) with a radius \( R \):

• \( |x - a| < R \)

However, there are special cases:
If the series converges only at the center \( x = a \), the interval of convergence is a single point.
For our series, the Ratio Test told us the radius of convergence \( R \) is 0.
The series only converges when \( x = 0 \), forming the interval \([0, 0]\). This narrow interval means the set of \( x \) values is merely a point, illustrating that outside this, the series behaves poorly and diverges.